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On pseudo-effect algebras which can be covered by pseudo MV-algebras

Dvurečenskij A., Vetterlein T.

Demonstratio Mathematica

2003

Vol. 36, nr 2

261--282

Pseudo-effect algebras are partial algebras (E; +, 0,1) which were recently introduced. They have a partially defined addition + which is only associative and not necessary commutative and with two complements, left and right ones. They are a non-commutative generalization of orthomodular posets and MV-algebras, respectively. We define five kinds of compatibilities, and we introduce a block as a maximal set of mutually compatible elements. The compatibility is a property of the physical system which corresponds to the distributivity, or equivalently, to "classical mechanics"-type phenomena. We show that any lattice pseudo-effect algebra under a natural condition can be covered by blocks, and any block is a pseudo MV-algebra. This result generalizes the analogical result of Riecanova for effect algebras. If the pseudo-effect algebra with the condition is, in addition, a (7-complete lattice, then it is a commutative effect algebra which can be covered by cr-complete MV-algebras.